2012-04-17

Generalized SDOF Systems

Sometimes an articulated rigid body happens to be a single degree of freedom system, sometimes we want to approximate the dynamical behaviour of a deformable, infinite degrees of freedom system with a simple, single degree of freedom model.

First, we study the behaviour of assemblages of rigid bodies, connected by bilateral, frictionless constraints and by springs and dampers, when the resulting system has one degree of freedom. Under the hypotesis of small displacements we find the instantaneous motion of every piece of the system, and write the equation of equilibrium (including inertial forces) using the principle of virtual displacements. The concept of generalized characteristic is introduced.

A similar procedure can be applied also to deformable bodies, assuming an instantaneous motion that satisfies external constraints and taking into account the internal virtual work in writing the equation of motion. Solving the equation of motion gives the amplitude of the assumed shape motion of the deformable body.

The procedure can be formalized to the so called "Rayleigh method" for computing an estimate of the first mode of vibration of, and we demonstrate that Rayleigh estimates have a lower bound in the true frequency of vibration of the system.

Good estimates require an assumed shape of vibration that's a good approximation of the true shape of vibration; we will see what discriminates a good from a bad shape, that is respect of the external constraints and a realistic distribution of internal stresses.

Starting from the above discussion of good shapes, we formalize an iterative method to obtain better and better estimates of the frequency of vibration.

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Giacomo Boffi